WASHINGTON (CNN) -- President Barack Obama claimed the second major legislative victory of his young administration Wednesday, signing a bill to provide federally funded health care to an estimated 4 million children.

President Obama says the SCHIP bill is a downpayment on his "commitment to cover every single American."

The final version of the new law, which expands the State Children's Health Insurance Program (SCHIP) by roughly $35 billion over the next five years, passed a sharply polarized House of Representatives earlier in the day, with almost every Democrat voting in favor of the expansion and most Republicans opposing it.

With the bill, Obama said at a White House ceremony, "We fulfill one of the highest responsibilities that we have -- to ensure the health and well-being of our nation's children."

The president said the bill was a downpayment on his "commitment to cover every single American."

The SCHIP expansion is Obama's second major legislative win in less than a week. The first was Thursday's approval of the Lilly Ledbetter Pay Equity Act, which makes it easier to sue employers for wage-based discrimination. Learn more about the SCHIP program »

The expansion is also a sign of the strength of Washington's new Democratic majority. Former President George W. Bush vetoed two similar health care bills in 2007, arguing that the legislation would encourage families to leave the private insurance market for the federally funded, state-run program.

Before the bill's passage, SCHIP covered almost 7 million children whose parents earn too much to qualify for Medicaid -- the federal health insurance program for the poor -- but who can't afford private insurance.

The new law boosts total SCHIP funding to approximately $60 billion. The expanded program will be financed with a 62-cent-per-pack increase in the federal tax on cigarettes.

"This is a day worthy of celebration. There can be no greater cause ... than protecting the well-being of our nation's children," New Jersey Rep. Frank Pallone, the legislation's primary House author, said shortly before the bill's final passage on a 290-135 vote.

Passing the health program's expansion is "morally the right thing to do by our children," said freshman Rep. Tom Perriello, D-Virginia. "At a time when the cost of health care is crushing America's families ... this is an important lifeline."

Opponents of the legislation argued that, among other things, it will allow undocumented immigrants to illegally access taxpayer-financed health care, and is insufficiently funded.

"This will go out of control just like all the other [entitlement] programs have, and our children will pay," Rep. Jack Linder, R-Georgia, warned during the House debate Wednesday.

Rep. Steve King, R-Iowa, ripped the bill as a "foundation stone for socialized medicine in the United States," arguing that raising the income limit for SCHIP eligibility will serve as the basis for a massive expansion of government-run health care.

The Senate passed the expansion Friday in a 66-32 vote. All those voting against the bill were Republicans, though nine Republicans voted in favor of the measure.